Exact solvability and unified analytical treatments to qubit-oscillator system

1. 《第六届全国冷原子物理和量子信息青年学者学术讨论会》《第六届全国冷原子物理和量子信息青年学者学术讨论会》 2012年8月14日-18日, 浙江师范大学 Exact solvability and unified analytical treatments to qubit-oscillator system Qing-Hu Chen (陈 庆 虎) Center for Statistical and Theoretical Condensed MatterPhysics, Zhejiang Normal University, Jinhua 321004, China & Department of Physics,Zhejiang University, Hangzhou 310027, China arXiv: 1204.3668, Phys. Rev. A, in press arXiv: 1204.0953 arXiv: 1203.2410 本人 1966 年出生， 早就不属于青年学者 向青年朋友请教了

2. Collaborators Prof. Ke-Lin Wang Department of Modern Physics, University of Science and Technology of China, Hefei 230026 Chen Wang (Ph. D student) Department of Physics, Zhejiang University, Hangzhou 310027 Dr. Yu-Yu Zhang (Former Ph. D student) Center for Modern Physics, Chongqing University， Congqing 400044 Shu He ( MS student), Prof. Tao Liu School of Science, Southwest University of Science and Technology, Mianyang 621010 Prof. Shi-Yao Zhu Beijing Computational Science Research Center, Beijing 100084

3. Brief introduction to quantum Rabi model (QRM) Rabi, Phys. Rev. 49, 324 (1936); 51, 652 (1937). □The interaction of two-level atom (qubit) with a bosonic mode ωis the resonant frequency of the cavity, Δ is the the transition frequency of the qubit, and g is the coupling strength, σx,z is usual Pauli matrix, a(a+) is the boson annihilation (creation) operator. δ=Δ- ω is the detuning. quantum Rabi model (Cavity QED) qubit-oscillator system (Circuit QED) In the fully quantum mechanical version Analytically unsolvable ! □Jaynes-Cummings (JC) model (1963) under the rotating-wave approximation (RWA) is analytically solvable. The counter rotating terms (CRTs) is omitted RWA CRTs

4. The Rabi model (RM) describes the simplest interaction between light and matter. Although this model has had an impressive impact on many fields of physics ---many physicists may be surprised to know that the quantum Rabi model has never been solved exactly. In other words, it has not been possible to write a closed-form, analytical solution for it

5. Outline 1. Exact solution for Qubit-Oscillator Systems (1) Numerically exact (2) Analytically exact (3) Applications No explicitly expression 2. Unified analytical treatments to qubit-oscillator systems explicitly expression but complicated 3. Concise first-order corrections to the RWA explicitly expression but very simple

6. Part I, Exact solution to the Quantum Rabi model (QRM) □ In RWA, the n-th eigenstate is at resonance, δ=0 The ground-state

7. Vacuum Rabi splitting in the JC model Measured transmission spectrum showing the vacuum Rabi mode splitting The atom is excited by the operator spontaneous emission to GS state The emission spectrum has two peaks with equal height (the distance of the two peaks, 2g, is the vacuum Rabi splitting). 2g is the energy difference of the 1st and 2nd eigenstates Wallraff et al., Nature 431, 162(2004).

8. The collapses and revivals in the evolution of the atomic population inversion If initially in Photonic Fock state |e, n> This is the quantum Rabi oscillation. If initially in photonic coherent state Population inversion under RWA can be evaluated analytically M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, 1997

9. revivals collapses <σz(t)>

10. Strong coupling Qubit-Oscillator System Deppe et al., Nature physics 4, 686(2008) Circuit quantum electrodynamics (QED) system

11. Experiments: T. Niemczyk et al., Nature Physics 6, 772 (2010) Title: Beyond the Jaynes-Cummings model: circuit QED in the ultrastrong coupling regime FIG. 1: Quantum circuit and experimental setup.

12. P. Forn-Diaz and J. E. Mooij et al., PRL105, 237001 (2010). arXiv: 1005.1559. Spectrum of the flux qubit coupled to the LC resonator.

14. Theory of Qubit-Oscillator Systems (Biased QRM) QHC, Tao Liu, and Kelin Wang, arXiv: 1007.1747, Chin. Phys. Lett. 29, 014208 (2012) The flux qubit behaves effectively as a two-level system Δ is the tunnel coupling The model Hamiltonian can be expressed as ωq is the atomic Larmor frequency, ω is the cavity frequency. g is the qubit-resonator coupling strength, enhanced by Josephson junction inductance Circuit QED Cavity QED: g/ω ~ 10-6 10-3 0.01 0.1 Nature 431, 162 (2004). Nature Physics 4, 686 (2008) Nature Physics 6, 772 (2010).

15. Numerically exact solution to QRM for ε≠0 (δΦ≠0) QHC et al, arXiv: 1007.1747 Transformation (ω=1) Ansatz for the wavefunction Laguerre polynomial ◇For strong coupling or highly excited states, much better than exact diagonalization in a-space

16. The optimum fitted parameters of the experimental results □The theoretical results are in good agreement with the experimental observations

17. Numerically exact solution to unbiased QRM: ε=0 QHC et al, PRA 82, 052306(2010) Hamiltonian Parity opertor The system is of even (+) or odd (-) parity. dn=±cn The wavefunction is reduced to S-equation The level transition is only allowed between the even and odd parity

18. First application to the entanglement dynamics □ Entanglement : highly nonlocal— shared among pairs of atoms, photons, electrons, etc., they may be remotely located and not interacting with each other. □ Entanglement as a resource in new approaches to both computation and communication Yu and Eberly Science 2009

19. CAB in two identical JC atoms without RWA □ Initiated from Bell state 1, ESD appear in non-RWA; disappear in RWA □ No periodicity of entanglement evolution for large g (a) Bell state 1 for α=π/4 (b) Bell state 2 for α=π/12

20. Effect of photonic number on ESD □CAB and Nph opposite behavior □ Nph suppress CAB Bell state 1 Possible origin of ESD Bell state 2

21. Our analytically exact solution to the Rabi model QHC, Tao Liu, Yu-Yu Zhang, and Ke-Lin Wang, EPL 96, 14003 (2011) arXiv: 1011.3280 The original JC model Parity To solve it easily, we introduce

22. One dimensional large Frolich polarons Qing-Hu Chen, Kelin Wang, and Shaolong Wan, J. Phys.: Condens. Matter 6, 6599(1994) The wavefunction Ntr=3 Ntr=2 where to find the solution for α(q)

23. Ntr=2 Ntr=3

24. The wavefunction Eigenstate of the pairty: +1-> even parity - 1-> odd parity Schroedinger equation

25. Analytical solution Comparing with the coefficients Without normalization, we can set c0=1 The linear term in a+ in the Fock space can be determined by eigenvalue of the pure coherent state exp(αa+ )|0>, we can set c1=0 energy Recurrence equation

26. Analytical solution can be expressed by α, Then we have (g, ∆,ωare given model parameters) A polynomial equation for only one variable α, only real value of αis reasonable. Zeros of f(α) give α Eigenenergies Eeigenfunctions

27. First order approximation (Ntr=2) Ground-state energy Irish PRL07: Generalized RWA, ED: exact diagonalization in Bosonic Fock space ◇ Much better than GRWA method

28. Full solutions y=f(α) odd parity, Ntr=60 Even parity, Ntr=59

29. convergence

30. Remarks on QHC et al, EPL 96, 14003 (2011) arXiv: 1011.3280 The JC model without the RWA can be mapped to a polynomial equation with a single variable. Its solutions recover exactly all eigenvalues and eigenfunctions of the model for all coupling strengths and detunings. In the past 80 years, it is analytically unsolvable. 4 months later, what happen?

31. See also arXiv: 1103.2461

32. E. Solano, Physics 4, 68 (2011).

33. Supplemental Material to Braak PRL paper use the representation of bosonic creation and anihilation operators in the Bargmann space of analytical functions in a complex variable z R. Koc et al., J. Phys. A: Math. Gen. 35, 9425(2002). transcendental function Zeros of the G-function gives all eigenenergies with parity ±1

34. arXiv:1204.3856 A recent work by Braak has renewed the interest in the old problem of coupling a photon field to a single spin 1/2 state, using the Rabi model. The central statement of this work is that the eigenfunctions in Bargmann representation must be analytic functions in the entire complex plane. Based on this condition, a procedure is derived from the series expansion of the eigenstates which provides a recursive evaluation of the spectrum. … In the following, it is shown that the use of the extra condition of analyticity of the eigenfunction in Bargmann representation is not necessary. Travenec: PRA 85, 043805 (2012) Nevertheless, there are disputes on whether the term exact solvability should be used, if the G functions are given only by some Taylor expansions with coefficients coming from a recurrence scheme. In my opinion, the word integrability should be left rather for models where a sufficient number of integrals of motion are known, which is not the case for Rabi models.

35. Exact solvability of the quantum Rabi models within Bogoliubov operators QHC et al., Phys. Rev. A ( in press, 2012), see also arXiv:1204.3668 A proof of our forthcoming article

36. Qing-Hu Chen et al., Phys. Rev. A ( in press, 2012) 1. Re-derivation of Braak's solution in a physical way! two Bogoliubov transformations

38. If both wavefunctions are the true eigen-function for a non-degenerate eigenstate with eigenvalue E, they should be in principle only different by a complex constant r where

39. Final G- function Braak, PRL 2011 ε=0

40. □Within extended coherent states, a recent exact solution to the quantum Rabi model [Daniel Braak, Phys. Rev. Lett. 107, 100401(2011)] can be recovered in an alternative simpler and more physical way, without uses of any extra conditions. Dear Qinghu, Many thanks for your interesting paper on the derivation of the G-function for the generalized Rabi model. Your method is certainly simpler than my approach, which was based on symmetry considerations,...... The question is: how do you justify this condition of proportionality which in turn gives the spectrum? In the Bargmann space approach it is the condition of analyticity throughout the whole complex plane which determines whether a state is an element of the Hilbert space or not - and this leads together with Z_2-symmetry to the G-function. What corresponds to the Bargmann condition in your approach? . Best regards, Daniel the extra condition is covered in the vacuum state in the space of the Bogoliubov operators. These vacuum states are well defined and known as the coherent states, so the present derivation is more physical and simpler.

41. The Juddian solutions originate from the properties of degeneracy □ Both Hilbert spaces in the two Bogoliubov operators are complete, if truncation is not done, the proportionality is justified naturally for non-degenerate states. □Koc et al in J. Phys. A: Math.Gen. 35, 9425(2002)have obtained isolated exact solutions in the QRM, which are just the Juddian solutions with doubly degenerate eigenvalues. The degenerate eigenstates areexcluded in principle in the solutions based on the proportionality.It naturally follows that the Juddian solutions are exceptional ones is not analytic in x but has simple poles at x=0,1,2…. All exceptional eigenvalues have the form En=n-g2 the necessary and sufficient condition for the occurrence of this eigenvalue is fn(x)=0

42. Comparisons with our previous work I □Qing-Hu Chen, Tao Liu, Yuan Yang, and Kelin Wang, PRA 82, 052306(2010) Qing-Hu Chen, Lei Li, Tao Liu, and Kelin Wang, arXiv: 1007.1747 Braak, PRL 107, 100401(2011). His wavefunctions are unfold by us link coefficients in two ansatz of the wavefunction ◇No essential differences, except that the avenues to obtain the basically same coefficients are different !

43. Comparisons with our previous work II □Qing-Hu Chen, Tao Liu, Yu-Yu Zhang, and Ke-Lin Wang, EPL 96, 14003 (2011) arXiv:1011.3280 Braak’s G function Reduced to ◇zeros of the both functions defined through different power series can give the exact eigenvalues. Both are analytically exact solutions ◇In our practical evaluation, it is not more difficult to locate the zeros for our function than those proposed by Braak, because the poles at x = n emerging in the latter are not present in our earlier solution.

44. Both are not analytical closed-form solutions! □Braak’s solution the eigenvalues are given by the zeros of the above Heun functions, which can not be obtained without truncation in the power series. ◇the expansion can not be closed naturally like in the JC model under the RWA □Enrique Solano called on in viewpoint Physics 4, 68 (2011): An intense dialogue between mathematics and physics will be needed to describe and predict unprecedented physical phenomena, since they might be hidden in the quantum numbers associated with Braak’s integrability criterion and analytical expressions. Otherwise, the present achievement will remain but a mathematical monologue. ◇by our two theories, we can also describe any physical phenomena based on Braak’s solution. Until now, no exceptions!

45. Wolf, Kollar, and Braak, PRA 85, 053817 (2012), see also arXiv:1203.6039

46. Braak’s method is absolutely not the unique method to obtain their physical figures 4-7 By our two earlier methods, we can completely reproduce all their figures Ntr=36 It is funny that we have studied the dynamics much earlier, which is however still only on arXiv Yu-Yu Zhang, Qing-Hu Chen, Shi-Yao Zhu, arXiv: 1106.2191, Wolf, Kollar, and Braak, PRA 85, 053817 (2012), see also arXiv: 1203.6039

47. □Counter rotating effect on collapses and revivals (Yu-Yu Zhang, QHC, Shi-Yao Zhu, arXiv: 1106.2191 ) Population inversion i.e. <σz(t)> Evolution at different g

48. P(t) is periodic with2π， g=2， periodicity τ0=4gπ=25.132 the mismatched phase in cos function in the summation at other time , leading to collapses.

49. □ Instead of studying the evolution of < σx> , we studied the vacuum Rabi splitting (Yu-Yu Zhang, QHC, Shi-Yao Zhu, arXiv: 1106.2191 ) Emission spectrum for g=0.1 and 0.8 at resonance Different heights More than 2 peaks

50. V|GS> No splitting